A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy

In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov--Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized by a function of two variables. In particular, we show that there are no isolated soliton type waves in the large time asymptotics for these solutions in contrast with well-known large time asymptotics for solutions of the KdV equation with reflectionless initial data

Kink Dynamics in a Topological Phi^4 Lattice

It was recently proposed a novel discretization for nonlinear Klein-Gordon field theories in which the resulting lattice preserves the topological (Bogomol'nyi) lower bound on the kink energy and, as a consequence, has no Peierls-Nabarro barrier even for large spatial discretizations (h~1.0). It was then suggested that these ``topological discrete systems'' are a natural choice for the numerical study of continuum kink dynamics. Giving particular emphasis to the phi^4 theory, we numerically investigate kink-antikink scattering and breather formation in these topological lattices. Our results indicate that, even though these systems are quite accurate for studying free kinks in coarse lattices, for legitimate dynamical kink problems the accuracy is rather restricted to fine lattices (h~0.1). We suggest that this fact is related to the breaking of the Bogomol'nyi bound during the kink-antikink interaction, where the field profile loses its static property as required by the Bogomol'nyi argument. We conclude, therefore, that these lattices are not suitable for the study of more general kink dynamics, since a standard discretization is simpler and has effectively the same accuracy for such resolutions.Comment: RevTeX, 4 pages, 4 figures; Revised version, accepted to Physical Review E (Brief Reports

Leading Order Temporal Asymptotics of the Modified Non-Linear Schrodinger Equation: Solitonless Sector

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution equations (NLEEs) integrable in the sense of the inverse scattering method, we obtain, in the solitonless sector, the leading-order asymptotics as $t$ tends to plus and minus infinity of the solution to the Cauchy initial-value problem for the modified non-linear Schrodinger equation: also obtained are analogous results for two gauge-equivalent NLEEs; in particular, the derivative non-linear Schrodinger equation.Comment: 29 pages, 5 figures, LaTeX, revised version of the original submission, to be published in Inverse Problem

Surface polaritons on left-handed cylinders: A complex angular momentum analysis

We consider the scattering of electromagnetic waves by a left-handed cylinder -- i.e., by a cylinder fabricated from a left-handed material -- in the framework of complex angular momentum techniques. We discuss both the TE and TM theories. We emphasize more particularly the resonant aspects of the problem linked to the existence of surface polaritons. We prove that the long-lived resonant modes can be classified into distinct families, each family being generated by one surface polariton propagating close to the cylinder surface and we physically describe all the surface polaritons by providing, for each one, its dispersion relation and its damping. This can be realized by noting that each surface polariton corresponds to a particular Regge pole of the $S$ matrix of the cylinder. Moreover, for both polarizations, we find that there exists a particular surface polariton which corresponds, in the large-radius limit, to the surface polariton which is supported by the plane interface. There exists also an infinite family of surface polaritons of whispering-gallery type which have no analogs in the plane interface case and which are specific to left-handed materials.Comment: published version. v3: reference list correcte

Discrete breathers in classical spin lattices

Discrete breathers (nonlinear localised modes) have been shown to exist in various nonlinear Hamiltonian lattice systems. In the present paper we study the dynamics of classical spins interacting via Heisenberg exchange on spatial $d$-dimensional lattices (with and without the presence of single-ion anisotropy). We show that discrete breathers exist for cases when the continuum theory does not allow for their presence (easy-axis ferromagnets with anisotropic exchange and easy-plane ferromagnets). We prove the existence of localised excitations using the implicit function theorem and obtain necessary conditions for their existence. The most interesting case is the easy-plane one which yields excitations with locally tilted magnetisation. There is no continuum analogue for such a solution and there exists an energy threshold for it, which we have estimated analytically. We support our analytical results with numerical high-precision computations, including also a stability analysis for the excitations.Comment: 15 pages, 12 figure

Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of solutions of the Korteweg--de Vries equation which are decaying perturbations of a quasi-periodic finite-gap background solution. We compute a nonlinear dispersion relation and show that the $x/t$ plane splits into $g+1$ soliton regions which are interlaced by $g+1$ oscillatory regions, where $g+1$ is the number of spectral gaps. In the soliton regions the solution is asymptotically given by a number of solitons travelling on top of finite-gap solutions which are in the same isospectral class as the background solution. In the oscillatory region the solution can be described by a modulated finite-gap solution plus a decaying dispersive tail. The modulation is given by phase transition on the isospectral torus and is, together with the dispersive tail, explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve.Comment: 45 pages. arXiv admin note: substantial text overlap with arXiv:0705.034

Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.Comment: 31 page

phi^4 Kinks - Gradient Flow and Dynamics

The symmetric dynamics of two kinks and one antikink in classical (1+1)-dimensional $\phi^4$ theory is investigated. Gradient flow is used to construct a collective coordinate model of the system. The relationship between the discrete vibrational mode of a single kink, and the process of kink-antikink pair production is explored.Comment: 23 pages, LaTex, 11 eps figures. We have added some clarification of our metho